Smallest \(g\)-supersolution with constraint
This paper studies the backward stochastic differential equation \[ Y_t = \xi + \int_t^T g(s,Y_s,Z_s) ds + A_T-A_t - \int_t^T Z_s dW_s \] with the constraint \(\varphi(t,Y_t,Z_t)\equiv 0\). Assuming the existence of a solution satisfying additional integrability requirements, the author shows the existence of a unique minimal solution within this class. This extends results by \textit{S. Peng} [Probab. Theory Relat. Fields 113, No. 4, 473-499 (1999; Zbl 0953.60059)] to the case where \(g\) and \(\varphi\) are not Lipschitz, but only satisfy a Yamada-type condition; the argument relies on results by \textit{X. Mao} [Stochastic Processes Appl. 58, No. 2, 281-292 (1995; Zbl 0835.60049)]. If \(g^\alpha\) and \(\xi^\alpha\) depend in addition on a parameter \(\alpha\) and the essential suprema of \(g^\alpha\) and \(\xi^\alpha\) over \(\alpha\) again lie in this family, the solution to the problem with the maximal coefficients is shown to be the essential supremum of the solutions \(Y^\alpha\). The key result behind all this is an extension of the comparison theorem for BSDEs to the present non-Lipschitz case.
backward stochastic differential equations
scientific article; zbMATH DE number 1542570